As One Pleases, One's Own Convenience, One's Own Way, One's Way
noun, な adjective
The hand (or person, as you know 手 often refers to a person) that wins gets to do as one pleases without regard for anyone else.
かって
The reading for this is weird. It's all kun'yomi, which luckily you've seen before, but still strange. I guess when you do as one pleases you can read kanji however you want, in this case the kun'yomi readings. Do note that there's a small っ in there before 手, presumably to connect the two readings of the two kanji together.
ほんと、勝手だよね！
You are so selfish!
勝手なことばっかり言わないでよ！
Don't say whatever you want.
他人の家で勝手に冷蔵庫を開けたりポテチを食べるの、やめてくれる？。
Could you stop opening the fridge or eating chips at someone else's house without permission?